Hexagon-deletion conjecture for maximum distance sets

Let a triangular lattice be the set of points with coordinates $a(1,0)+b(\frac12,\frac{\sqrt3}{2})$, where $a,b\in\mathbb Z$. Consider a regular hexagon or an equiangular hexagon whose side lengths alternate between $k$ and $k+1$, together with all triangular-lattice points in that hexagon. Hexagon-deletion conjecture. For any $m\geq 3$, at least one construction of a maximum $m$-distance set can be realized by repeatedly removing points from these hexagons. The paper gives examples for $m=4$ and $m=6$, but does not establish the assertion for all $m$.